﻿ 因素空间中属性约简的区分函数
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 智能系统学报  2017, Vol. 12 Issue (6): 889-893  DOI: 10.11992/tis.201609014 0

### 引用本文

QU Guohua, LI Chunhua, ZHANG Qiang. Attribute reduction and discernibility function in factor space[J]. CAAI Transactions on Intelligent Systems, 2017, 12(6): 889-893. DOI: 10.11992/tis.201609014.

### 文章历史

1. 山西财经大学 管理科学与工程学院，山西 太原 030006;
2. 北京理工大学 管理与经济学院，北京 100081

Attribute reduction and discernibility function in factor space
QU Guohua1, LI Chunhua1, ZHANG Qiang2
1. School of Management Science and Engineering, Shanxi University of Finance and Economics, Taiyuan 030006, China;
2. School of Management and Economics, Beijing Institute of Technology, Beijing 100081, China
Abstract: To enable description, Rough Set theory uses an information system constructed by attributes, and various detailed entropy indexes are employed to achieve the scale of information; this provides a mathematical basis for knowledge mining of relational databases. Current research is focused on the role that Rough Set plays in attribute reduction; however, definition of the discernibility function used for attribute reduction is unclear. For example, when there is no attribute to distinguish between two objects, it is unclear why 1 is used instead of 0 for the corresponding attribute variable. As such, this problem causes a bottleneck when applied in Rough Set. The aim of this paper is to find a more reasonable explanation and application for discernibility functions. The method firstly defines the operation between attribute names, which is different from the operation between attribute values, and the attribute name is different from the attribute value. If operation of the attribute value is confused with that of the attribute name, the meaning will subsequently be unclear. To avoid such confusion, Factor Space theory is employed, as it treats attribute names as factors. The theory uses the operation between factors to define the operation of the attribute name, enabling clear definition of the discernibility function, and explains why the attribute variable takes the value of 1 under special circumstances. Results indicate that Factor Space theory can deepen the theoretical basis of Rough Set and improve its ability to solve problems.
Key words: factor space    rough set    factor reduction    discernibility function    factorial causality analysis

1 粗糙集信息系统

 $\alpha (x,y) = \left\{ {a \in A\left| {f(x,a) \ne f(y,a)} \right.} \right\}$ (1)

 $\Delta = \prod\nolimits_{\left( {x,y \in U \times U} \right)} {\left( {\sum {\alpha (x,y)} } \right)}$ (2)

2 因素空间中分辨过程的因素约简

1) 指标集 $F = ( \vee , \wedge {,^c}, {1}, {0})$ 是一个完全的布尔代数；

2) X(0)={∅}；

3) 对任意 $T \subseteq F$ ，若

 $(\forall s,t \in T)(s \ne t \Rightarrow s \wedge t = {\rm{0}})$

$X( \vee \{ f|f \in T\} ) = {\prod _{f \in T}}X(f)$

4) $\forall f \in F$ ，都有一个映射

 $f:D(f) \to X(f){\kern 1pt} {\kern 1pt} (D(f) \subseteq U)$

F叫做因素集，其最大、最小元1和0分别叫做全因素和零因素，X(f)叫做f-性态空间。

f(u, v)是能区分uv的因素的最大下级公共因素。这样的因素越少，公共下级因素就越大，当没有这样的因素时，公共下级因素就取最大。全因素应当能区分所有的对象。对角线上的元素全是1。这也解释了定义中的约定为什么是合理的。

 $d\left( A \right) = \vee \{ f\left( {u,v} \right)|u,v \in A\} = \vee \{ \wedge \{ f|f\left( u \right) \ne f\left( v \right)\} |u,v \in A\}$

g能分辨A，则对任意u, vA, uv, 都有g(u)≠g(v)。于是， $g \in \{ f|f\left( u \right) \ne f\left( v \right)\}$ 。所以f(u, v)≤g。由于这个不等式对任意u, vAuv）都成立，便有d(A)≤g，故d(A)是能分辨A的最小因素。

3 结束语

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